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Reaching the Limits of Enhancement in (sub)Nanometer Metals Structures Reuven Gordon, and Aftab Ahmed ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b01227 • Publication Date (Web): 24 Oct 2018 Downloaded from http://pubs.acs.org on October 25, 2018
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ACS Photonics
Reaching the Limits of Enhancement in (sub)Nanometer Metals Structures Reuven Gordon∗,†,¶ and Aftab Ahmed‡ Department Electrical and Computer Engineering, University of Victoria, Victoria, British Columbia V8P 5C2, Canada, and Department of Electrical and Computer Engineering, California State University, Long Beach, California 90840, USA E-mail:
[email protected] Phone: +1 250 472 5179 . Fax: +1 250 721 6052
Abstract Plasmonic enhancement has had remarkable success in optical coupling to the nanometer scale, enabling feats such as Raman spectroscopy with single molecule sensitivity. Here it is argued that much greater enhancements are possible in the near future by combining the gains of plasmonic resonances, directivity, sub-nanometer gaps and permittivity near zero materials. The pursuit of such extraordinary enhancements promises to bring new physics such as peering into the world of quantum optomechanics. It also promises new applications such as quantitative single molecule Raman spectroscopy and low photon number nonlinear optical switching. In addition, by pushing the limits of plasmonic enhancement, it is expected that the community will gain a greater appreciation of how physical phenomena such as non-locality, surface scattering and quantum tunneling each play a role in determining the ultimate performance. ∗
To whom correspondence should be addressed ECE ‡ ECE ¶ Center for Advanced Materials & Related Technologies (CAMTEC), University of Victoria, Victoria, British Columbia V8W 2Y2, Canada †
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Keywords Plasmonics, Raman, Nonlinear Optics, Near-field
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Introduction
It is well known that metal nanostructures can enhance the local electric field, and this phenomenon has been exploited in: surface-enhanced Raman scattering, 1–5 surface enhanced IR absorption, 6 nonlinear wavelength conversion, 7 multi-photon fluorescence, 8,9 optical tweezers 10 and quantum coupling. 11 While the basic theoretical understanding of plasmonic resonances in nanostructured metals has been present for more than a century 12 (see Figure 1), there are still several conflicting reports in the literature about the ultimate limits of plasmonic enhancement. For example, some works suggested that non-local effects would reduce the plasmonic enhancement and effectively increase the gap distances, whereas theoretical works have suggested that the gap distance is narrowed once again by the electron wavefunction entering the gap region. 13,14 Many works have considered field enhancements from singularities from sharp tapers and nearly touching metals in the absence of quantum or non-local effects. 15 As dimensions of fabrication enter the sub-nanometer regime, possibilities are emerging for greater field enhancements than previously thought possible with classical models. Field enhancement in metal nanostructures arises from three main sources: local geometry of the permittivity (gaps, lightening rod effects), resonant field build up (plasmonic resonances), and efficient coupling (directivity, adiabatic enhancement). Affecting this field enhancement are material losses, surface scattering, the single channel limit, the non-local material response, quantum effects (confinement and tunneling of electrons) and material damage (such as melting). Here we aim to provide a perspective on field enhancements, considering existing achievements and what is ultimately possible. The overall finding is that significantly greater enhancement is within reach, and even greater enhancements than 2
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Figure 1: Scattering of a gold nanoparticle showing plasmon resonance peak at around 540 nm, as compared with perfect conductor nanoparticles (dashed line includes higher order contribution). Horizontal axis is wavelength in nanometers. From Ref. 12 we originally thought possible.
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Enhancement Factors
The spheroid is a canonical example in the analysis of plasmonic effects. 16–19 Many other field enhancing geometries exist, and these may perform better in different wavelength regions. 20–22 The goal here, however, is to understand the different enhancement factors that come from nanostructuring and the material response. The local field enhancement with respect to the incident plane wave field from a small prolate spheroid at the end of the long axis can be expressed analytically as:
f=
m A(m − d ) + d −
i4π 2 V 3/2 (m 3λ3 d
− d )
(1)
where m,d is the relative permittivity of the metal, surrounding dielectric, V is the volume of the spheroid, λ is the free-space wavelength, and A is a geometric factor.
ξ+1 A = (ξ − 1)(ξ/2) ln −1 ξ−1 2
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(2)
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√ where ξ = a/ a2 − b2 and a and b are the long and short axes of the spheroid. From Eq. 1 it appears shrinking the particle will give higher field enhancement, up to the point where the permittivity is still valid. This will be discussed in more detail below.
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Plasmonic Enhancement and Local Geometry
Figure 2: (a) Ratio of real and imaginary part of permittivity for silver (solid) and gold (dashed). (b) Maximum field intensity enhancement for a spheroid for silver (solid) and gold (dashed). Data from set C of previous work. 23 The plasmonic resonance is defined as the wavelength where the real part of the denominator in Equation 1 is approaches zero. For a very small volume and d = 1, the field
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intensity enhancement may be found as: |f |2 =
0m 00 m
2
+1 (3)
A2
where 0m and 00m are the real and imaginary parts of the metal’s relative permittivity. This shows two important features: first that the magnitude of the ratio between the real and imaginary parts of the relative permittivity should be large (a common figure of merit), and second that A should be small (the lightning rod effect). Typical values of
0m 00 m
and 1/A
are around 3–30 in the visible and near-IR, so that 100–105 is the range for field intensity enhancement expected. Figure 2a shows the ratio of the real and imaginary of the relative permittivity for gold and silver. We show an example calculation with 104 enhancement for a gold spheroid operating near the optimal permittivity condition in the Supporting Information. We may also consider operating at longer wavelengths by elongating the spheroid. The resonance condition (neglecting retardation) is given when the real part of the denominator in Eq. 1 is zero, or A = d / (d −
